Effectiveness analysis and optimum design of the rotary regenerator for thermophotovoltaic (TPV) system

Abstract

The influence of the period of rotation on the effectiveness of the thermophotovoltaic (TPV) rotary regenerator was theoretically and experimentally investigated. It was found that the deviations of the theoretical results from the experimental ones decrease with the increase of the period of rotation. To the TPV system of 10 kW combustion power, the deviation is 3.5% when the rotation period is 3 s; while the deviation decreases to 1.5% when the rotation period increases to 15 s. The deviation could be mainly attributed to the cold and hot fluids carryover loss which was not considered in the model. With a new model taking account of the carryover loss established, the predicted results were greatly improved. Based on the modified model, the influence of geometrical parameters of rotary regenerator on the effectiveness was analyzed for TPV systems of various combustion power. The results demonstrate that the effectiveness increases with the increase of the rotary regenerator diameter and height, while fluid carryover loss increases at the same time, which weakens the impact of geometrical parameters.

Keywords

thermophotovoltaic (TPV) system / rotary regenerator / effectiveness / carryover loss

Introduction

The concept of thermophotovoltaic (TPV) system was introduced by Aigrain in the 1960s. The system converts the infrared radiation from radiator directly into electric energy through the PV cells. A typical TPV system consists of a combustion-radiator, TPV cell modules, an optical filter and auxiliary equipments [1,2]. The regenerator is designed to preheat combustion air by utilizing high-temperature exhaust gas, which is a key component to increase the total efficiency of the system [3]. In Ref. [4], the author calculated and analyzed the TPV system of a 10 kW combustion power. The results demonstrate that the radiant surface temperature and the chemical-radiant energy conversion efficiency can be greatly raised by utilizing regenerator. When the effectiveness of the regenerator is 75%, the average radiant surface temperature and the chemical-radiant energy conversion efficiency can reach 1523.8 K and 73.7% respectively. Since the TPV system requires compact configuration and has the problems of low system efficiency and high-temperature exhaust gas [5], the regenerator must have the characteristics of miniaturization and high effectiveness. Rotary regenerator is especially suitable for TPV systems compared with other regenerators since it has a larger heat transfer area per unit volume, a higher efficiency and is easier to be miniaturized.

The rotary regenerator is a thermal energy storage device in which heat is transferred from one fluid to the other through temporary storage in the form of sensible heat in the matrix. To design a high-performance and compact rotary regenerator, a suitable theoretical model must be established. Early models only considered the convection between fluid and matrix as well as energy exchange in fluids themselves, neglecting the influence of matrix conduction [6]. Therefore Hill and Willmott developed an analytical solution for the periodic steady state of negligible matrix conduction in the axial direction and infinite in the radial direction [7]. In 2001, Klein and Eigenberger put forward a numerical approximation method based on closed equations, but the extension of application was limited because the mass flow rate of cold and hot fluids on both sides of the regenerator must be equal [8]. Sphaier and Worek established ϵ-NTU correlation formulas to analyze the influence of various parameters on the effectiveness of rotary regenerator [9]. Nair and Verma made use of finite difference method to solve the rotary regenerator energy conservation equation [10].The regenerator effectiveness and the temperature distribution of matrix and fluid could be available in different mass flow rate of cold and hot fluids. Currently, few experimental studies have been conducted on small rotary regenerators applicable to the TPV system, except for Colangelo and de Risi who made relevant researches on the TPV system of 12 kW combustion power without giving in-depth analysis of the influence of the regenerator geometrical parameters and operating conditions on the effectiveness [11]. The purpose of this paper is to develop a model to analyze the influence of the rotation period, the rotary regenerator diameter and height on the effectiveness of TPV systems of various combustion power. And an experimental system was established to verify the theoretical model.

Model description

Physical model and assumptions

The influence of fluid turbulence on effectiveness was considered in the theoretical model [10], but fluid diffusion coefficient must be measured in experiments and the results indicated that fluid turbulence has little influence on effectiveness, so a new theoretical model ignoring the fluid turbulence was built. The schematic diagram of the rotary regenerator is shown in Fig. 1. The whole regenerator is divided into two zones by plane ABCD and cool (hot) fluid passes the left (right) side. The rotary matrix inside the regenerator forms the periodically solid phase flow from the hot fluid to cold fluid. Therefore, the matrix is heated and cooled alternately. In this process thermal energy can be indirectly transferred from hot fluid to cold fluid. Here hot fluid is high-temperature combustion exhaust gas in the TPV system at an inlet temperature T₁; cold fluid is the air to be preheated at an inlet temperature T₂. In addition, in order to facilitate the analysis and calculation, the following assumptions are made:

1) Thermal properties of the two fluids and matrix are constant.

2) The fluids are in counterflow and the matrix rotates on a fixed axis.

3) The conductivity of the matrix is negligible along the direction of rotation, finite along the fluid flow direction and infinite in the radial direction.

4)The convective heat transfer coefficients are constant along the fluid flow direction.

5)There is no intermixing of the two fluids.

Mathematical model

Since the matrix conduction is infinite in the radial direction and the fluid temperature is the same in this direction without temperature gradient, the whole temperature field can be represented by the two-dimensional coordinate in the axial and rotational directions. So the regenerator can be discretized by cutting along the line AD shown in Fig. 2. Due to the fact that the fluids are in counterflow, the coordinate is reversed in the axial direction. In Fig. 2, the cross (x) represents the dimensionless matrix temperature,

θ_r=(T_r-T₂)/(T₁-T₂);

the dot (·) represents the dimensionless fluid temperature,

θ_f=(T_f-T₂)/(T₁-T₂).

It is noteworthy that θ_r,h(i,1) at the leftmost side of the grid is equal to θ_r,c(M-i+1,S) at the rightmost side, and θ_r,h(i,R) is equal to θ_r,c(M-i+1,1) on the line BC.

Figure 3 illustrates the schematic diagram of element. The energy balance for a typical element on the hot fluid side may be written as

(1)C_{\mathrm{h}}{\displaystyle \frac{\mathrm{\Delta }\varphi }{{\psi }_{\mathrm{h}}}}[{\theta }_{f,h}(i,j)-{\theta }_{f,h}(i+\mathrm{1},j)]+k\left[(\mathrm{1}-ϵ){\displaystyle \frac{R^{\mathrm{2}}}{\mathrm{2}}}\mathrm{\Delta }\varphi \right]\times {\displaystyle \frac{[{\theta }_{\mathrm{r},\mathrm{h}}(i-\mathrm{1},j)+{\theta }_{r,h}(i-\mathrm{1},j+\mathrm{1})]/\mathrm{2}-[{\theta }_{\mathrm{r},\mathrm{h}}(i,j)+{\theta }_{\mathrm{r},\mathrm{h}}(i,j+\mathrm{1})]/\mathrm{2}}{\mathrm{\Delta }z}}-k\left[(\mathrm{1}-ϵ){\displaystyle \frac{R^{\mathrm{2}}}{\mathrm{2}}}\mathrm{\Delta }\varphi \right]\times {\displaystyle \frac{[{\theta }_{\mathrm{r},\mathrm{h}}(i,j)+{\theta }_{\mathrm{r},\mathrm{h}}(i,j+\mathrm{1})]/\mathrm{2}-[{\theta }_{\mathrm{r},\mathrm{h}}(i+\mathrm{1},j)+{\theta }_{\mathrm{r},\mathrm{h}}(i+\mathrm{1},j+\mathrm{1})]/\mathrm{2}}{\mathrm{\Delta }z}}＝C_{\mathrm{r}}{\displaystyle \frac{\mathrm{\Delta }z}{L}}[{\theta }_{\mathrm{r},\mathrm{h}}(i,j+\mathrm{1})-{\theta }_{\mathrm{r},\mathrm{h}}(i,j)].

The first term in Eq. (1) is the heat transfer between hot fluid and matrix in the element. The second term is energy in by matrix conduction in the axial direction. The third term is energy out by matrix conduction in the axial direction, and the fourth term is energy stored in the element.

As for hot fluid, the energy conservation equation of the element can be written as

(2){(hA)}_{\mathrm{h}}{\displaystyle \frac{\mathrm{\Delta }\varphi }{{\psi }_{\mathrm{h}}}}\cdot {\displaystyle \frac{\mathrm{\Delta }z}{L}}{(\mathrm{\Delta }T)}_{\mathrm{a}\mathrm{v}\mathrm{g}}=C_{\mathrm{h}}{\displaystyle \frac{\mathrm{\Delta }\varphi }{{\psi }_{\mathrm{h}}}}[{\theta }_{\mathrm{f},\mathrm{h}}(i+\mathrm{1},j)-{\theta }_{\mathrm{f},\mathrm{h}}(i,j)].

where the left side of the equation indicates the convection between hot fluid and matrix in element, and {(\mathrm{\Delta }T)}_{\mathrm{a}\mathrm{v}\mathrm{g}} is the average temperature difference between the fluid and the matrix in the element,

{(\mathrm{\Delta }T)}_{\mathrm{a}\mathrm{v}\mathrm{g}}=[{\displaystyle \frac{{\theta }_{\mathrm{f},\mathrm{h}}(i,j)+{\theta }_{\mathrm{f},\mathrm{h}}(i+\mathrm{1},j)}{\mathrm{2}}}-{\displaystyle \frac{{\theta }_{\mathrm{r},\mathrm{h}}(i,j)+{\theta }_{\mathrm{r},\mathrm{h}}(i,j+\mathrm{1})}{\mathrm{2}}}].

The right side indicates the energy loss rate when hot fluid flows through the element.

The boundary condition at the entrance of the hot fluid side is written as

(3){\theta }_{\mathrm{f},\mathrm{h}}(\mathrm{1},j)=\mathrm{1.}

At the exit, the boundary conditions are

(4){\displaystyle \frac{\mathrm{d}{\theta }_{\mathrm{f},\mathrm{h}}}{\mathrm{d}z}}=\mathrm{0},

(5){\displaystyle \frac{\mathrm{d}{\theta }_{\mathrm{r},\mathrm{h}}}{\mathrm{d}z}}=\mathrm{0.}

Similarly, the heat transfer analysis process can be made for the cold fluid and matrix, only with a different boundary condition, θ_f,c(1,j)=0.

Computational procedure

1) Take the boundary conditions into Eqss. (1) and (2), respectively, then the temperature expressions of matrix as well as cold and hot fluids can be obtained row by row.

2) Assume an initial temperatureθ_r,h(i,1), the temperatures of matrix and fluid at the hot fluid sidecan be obtained from left to right. Then let

{\theta }_{\mathrm{r},\mathrm{h}}(i,R)={\theta }_{\mathrm{r}\mathrm{c}}(M-i+\mathrm{1},\mathrm{1})

repeat the above calculation process to obtain the temperatures of matrix and fluid at cold fluid side. If

it means the results are convergent. However, if the error is out of the specified value, then let

{\theta }_{\mathrm{r},\mathrm{h}}(i,\mathrm{1})-{\theta }_{\mathrm{r},\mathrm{c}}(M-i+\mathrm{1},S)

as a new initial temperature for recalculation until the results meet the error requirement. Finally, the temperature of matrix and fluids at both sides can be obtained and the rotary regenerator effectiveness can be calculated as

\eta ={\displaystyle \frac{{\left[{\theta }_{\mathrm{f},\mathrm{c}}\left(M+1,j\right)\right]}_{\mathrm{a}\mathrm{v}\mathrm{g}}-0}{1.0-0}}={\left[{\theta }_{\mathrm{f},\mathrm{c}}\left(M+1,j\right)\right]}_{avg}

Experimental description

Figure 4 represents the experimental system of the rotary regenerator. The whole system consists of four parts: rotary regenerator, speed governing system, high-temperature heating furnace and measurement system. The regenerator diameter and height are 120 and 76 mm. SC type corrugated stainless steel mesh is selected as matrix, which has a larger heat exchange area per unit volume (700 m²/m³) among the same type of products. The specific heat capacityβof that materialβis 0.5 kJ/(kg·K), the density and the porosity are 7817 kg/m³ and 0.87, respectively. In the experiment, air is heated by a high-temperature heating furnace whose maximum temperature can reach 1373 K to simulate high-temperature exhaust gas produced by the combustorββof TPV systems. Then the heated air (hot fluid) flows into the rotary regenerator and transfers heat with cold fluid indirectly. The rotary speed of the regenerator is regulated by changing the power of the drive motor with the governing box. The inlet mass flow rates of cold and hot fluids are controlled by two rotameters with a precision of±2.5%. Four thermocouples are placed in the inlets/outlets of cold and hot fluids. Except a T-type thermocouple with a precision of±1.0°C at the cold fluid entrance, the rest are K-type thermocouples with a precision of±0.4%. The data acquisition system is used to collect temperature data at an interval of 1 s. Experiments are designed to simulate the operating conditions of 5 and 10 kW combustion power, and test the effectiveness of the regenerator of rotation period of 3-15 s. The experimental operating conditions are listed in Table 1.

Results and discussion

Comparison of the experimental and theoretical results

The relationships between effectiveness and rotation period for different combustion powers are presented in Fig. 5. The rotary regenerator diameter and height are 120 mm and 76 mm respectively. As can be seen from the results, theoretical values of the effectiveness decrease with the increase of the rotation period, while the experimental values change consistently with the theoretical ones, but the deviations of the theoretical results from the experimental ones decrease. When the rotation period is 3-15 s and the combustion power is 5 kW, the deviation is 1.3%-3.9%, while the deviation is 1.5%-3.5% for 10 kW combustion power. The deviation could be attributed to bypass leakage and carryover loss not considered in the model. As there is gap between matrix and shell of the regenerator, some cold and hot fluids directly flow to their own exits from the entrances, namely bypass leakage. At the exit this part of leakage fluids mix with other fluids. As shown in Fig. 6, it makes the actual outlet temperature of cold/hot fluid lower/higher than the corresponding theoretical values. So the measured effectiveness of the regenerator is lower than the corresponding predicted result. In addition, there is some unavoidable carryover of a fraction fluid after the switching of the fluids through the same flow passages. This phenomena results in a cold fluid temperature increase and a hot fluid temperature decrease, so the heat transfer between the matrix and fluids is weakened and the effectiveness decreases. Under the same geometrical parameters and operating conditions, bypass leakage can be considered as a constant, so it can be deduced that the shorter the rotation period is, the more significant the carryover loss will be, and the greater influence it will have on the effectiveness. Based on the above analysis, the rotary regenerator should be improved by setting a sealing plate between the matrix and the shell to reduce bypass leak; and by setting a transition regionβwithout air flowingβbetween cold and hot fluids zone to segregates the cold and hot fluids to minimize the carryover loss.

Model improvement

The above analysis shows that the carryover loss is the main reason for the error between theoretical and experimental values, so the model is modified, namely, add “fluid carryover loss”, i.e.,

C_{\mathrm{l}}{\displaystyle \frac{\mathrm{\Delta }z}{L}}\left[{\displaystyle \frac{{\theta }_{\mathrm{f},\mathrm{h}}(i,j)+{\theta }_{\mathrm{f},\mathrm{h}}(i+\mathrm{1},j)}{\mathrm{2}}}-{\displaystyle \frac{{\theta }_{\mathrm{f},\mathrm{h}}(i,j-\mathrm{1})+{\theta }_{\mathrm{f},\mathrm{h}}(i+\mathrm{1},j-\mathrm{1})}{\mathrm{2}}}\right]

to the right of Eqs. (1) and (2). Here

C_{\mathrm{l}}={\displaystyle \frac{c_{\mathrm{p}}\mathrm{\pi }{R}^{\mathrm{2}}ϵL\rho }{\mathrm{2}\omega }}.

C_{\mathrm{l}}{\displaystyle \frac{\mathrm{\Delta }z}{L}}{\displaystyle \frac{{\theta }_{\mathrm{f},\mathrm{h}}(i,j)+{\theta }_{\mathrm{f},\mathrm{h}}(i+\mathrm{1},j)}{\mathrm{2}}}

indicates the energy flowing into the element carried by the fluid,

C_{\mathrm{l}}{\displaystyle \frac{\mathrm{\Delta }z}{L}}{\displaystyle \frac{{\theta }_{\mathrm{f},\mathrm{h}}(i,j-\mathrm{1})+{\theta }_{\mathrm{f},\mathrm{h}}(i+\mathrm{1},j-\mathrm{1})}{\mathrm{2}}}

indicates the energy leaving away from the element carried by the fluid.

As can be seen from Fig. 7, the modified theoretical results have been greatly improved. When the rotation period is 3-15 s and the combustion power is 5 kW, the deviation is 1.4%-1.7%, while the deviation is 1.1%-1.4% when the combustion power is 10 kW. Therefore, the modified model is adopted to analyze the influence of the regenerator diameter and height on the effectiveness.

Influence of regenerator geometrical parameters on the effectiveness

Figure 8 represents the influence of regenerator diameter on the effectiveness in TPV systems of various combustion power when the rotary regenerator height and rotation period are 76 mm and 3 s respectively. Figure 9 demonstrates the influence of regenerator height on the effectiveness when the diameter is 120 mm and the rotation period is 3 s. In Figs. 8 and 9, the solid symbols represent the results of the original model, while the hollow symbols represent the modified results. It can be seen clearly that the effectiveness increases with the increase of the height and the diameter for both the original and modified models, but the modified values are less than original ones. According to the classic theory of Kays and London, the effectiveness of rotary regenerator has a close relationship with three dimensionless parameters: NTU, C_r/_Cmin and C_max/C_min. In this paper, C_max＝C_h, C_min＝C_c. Under the same operating conditions, C_h/C_c is a constant. The heat exchange area, C_r/C_c and NTU increases with the increase of the height and diameter of the regenerator, thereby increasing the effectiveness of the regenerator. It is noteworthy that in the original model the effectiveness of the regenerator has a linear relationship with the diameter. This is because at present most theoretical models assume the matrix conductivity as infinite in the radial direction in order to achieve the numerical solution, resulting in the same matrix temperature in radial direction, so those models cannot accurately reflect the influence of the diameter on the effectiveness. According to Figs. 8 and 9, fluid carryover loss being considered in the modified model weakens the influences of the regenerator diameter and height on the effectiveness. With the increase of the diameter and height, the fluid carryover loss also increases and the effectiveness growth rate decreases. There is no more linear correlation between the diameter and effectiveness.

Conclusions

A theoretical model considering the matrix conduction in the axial direction was set up to analyze the influence of the rotation period on the effectiveness in two TPV systems of 5 and 10 kW combustion power respectively. The regenerators under the two operating conditions are experimentally simulated, and the results demonstrate that the maximum deviation between experimental values and theoretical ones are 3.9% and 3.5%, mainly due to the bypass leakage and carryover loss. So a modified model was presented and the accuracy was greatly improved. The maximum errors are less than 1.7% and 1.4% in the TPV system of 5 and 10 kW combustion power. Using the modified model, the influences of regenerator geometrical parameters on the effectiveness in three TPV systems of 5, 10 and 15 kW combustion power were analyzed. The results indicate that the effectiveness increases with the increase of the regenerator diameter and height, but the influences of the geometrical parameters on the effectiveness are weakened by the carryover loss. To a certain extent, the modified model improves the limitation of the assumption that matrix conductivity is infinite in radial direction.

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